Finally let us go back to the question \((\heartsuit )\)  appeared in the previous section. It is constructed in [1]} that for any sufficiently small \(\epsilon >0\)  there exists a sequence of complete Riemannian metrics \(g_i\)  on \(\mathbb {R}^8\)  with positive Ricci curvature such that
\(\left(\mathbb {R}^8, \mathsf {d}_{g_i}, 0, \frac{\mathrm {vol}_{g_i}}{\mathrm {vol}_{g_i}B_1(0)} \right) \stackrel{\mathrm {pmGH}}{\longrightarrow } \big (\mathbb {R}_{\ge 0} \times \mathbb {S}^4, \mathsf {d}_{h_{\epsilon }}, p, \nu _{\epsilon }\big ), \qquad h_{\epsilon }=(\mathop {}\!\mathrm {d}t)^2 +\left(\frac{t^{1+\epsilon }}{2}\right)^2g_{\mathbb {S}^4}\) 
